On a Recurrence Arising in Graph Compression

In a recently proposed graphical compression algorithm [1], the following tree arose in the course of the analysis. The root contains n balls that are consequently distributed between two subtrees according to a simple rule: In each step, all balls independently move down to the left subtree (say with probability p) or the right subtree (with probability 1−p). A new node is created as long as there is at least one ball in that node. Furthermore, a nonnegative integer d is given, and at level d or greater one ball is removed from the leftmost node before the balls move down to the next level. These steps are repeated until all balls are removed (i.e., after n + d steps). Observe that when d = ∞ the above tree can be modeled as a trie that stores n independent sequences generated by a memoryless source with parameter p. Therefore, we coin the name (n, d)-tries for the tree just described, and to which we often refer simply as d-tries. Parameters of such a tree (e.g., path length, depth, size) are determined by an interesting two-dimensional recurrence (in terms of n and d) that – to the best of our knowledge – was not analyzed before. We study it, and show how much parameters of such a d-trie differ from the corresponding parameters of regular tries. We use methods of analytic algorithmics, from Mellin transforms to analytic poissonization.